The doping of colloidal semiconductor nanocrystals (NCs) presents an additional knob beyond size and shape for controlling the electronic properties. An important problem for doping with aliovalent elements is associated with resolving the location of the dopant and its structural surrounding within small NCs, an issue directly connected with self-purification. Here we used a postsynthesis diffusion-based doping method for introducing Cu impurities into InAs quantum dots. X-ray absorption fine structure (XAFS) spectroscopy experiments along with first-principle density functional theory (DFT) calculations were used to probe the impurity sites. The concentration dependence was investigated for a wide range of doping levels, helping to derive a self-consistent picture where the Cu impurity occupies an interstitial site within the InAs lattice. Moreover, at extremely high doping levels, Cu-Cu interactions are identified in the XAFS data. This structural model is supported by X-ray diffraction data, along with the DFT calculation. These findings establish the reproducibility of the diffusion-based doping strategy giving rise to new opportunities of correlating the structural details with emerging electronic properties in heavily doped NCs.